On the Interactive Communication Cost of the Distributed Nearest Lattice Point Problem

01/30/2018
by   V. A. Vaishampayan, et al.
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We consider the problem of distributed computation of the nearest lattice point for a two dimensional lattice. An interactive two-party model of communication is considered. Algorithms with bounded, as well as unbounded, number of rounds of communication are considered. For the algorithm with a bounded number of rounds, expressions are derived for the error probability as a function of the total number of communicated bits. We observe that the error exponent depends on the lattice. With an infinite number of allowed communication rounds, the average cost of achieving zero error probability is shown to be finite.

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I Previous Work

The problem considered here is related to the following bodies of prior work: interactive communication, distributed function computation, distributed hypothesis testing and quantization, in particular, asymptotic quantization theory. We briefly review prior work in each of these areas. Loosely speaking, communication complexity is the minimum amount of communication required to achieve a specific objective, whether it be distributed compression or distributed function computation.

Two-party interactive communication is considered in a series of papers [25],[26][27]

. When worst-case complexity is considered, infinite message complexity can be as small, but no better than, the logarithm of the one-message complexity, and the one-message complexity is the logarithm of the strong chromatic number of a graph that is derived from the support set of the joint distribution of the pair of random variables. It is also shown that two messages suffice to achieve communication within a constant factor of the best possible using an infinite number of messages. For the average case, when random variables are uniformly distributed over their support set, average case communication close to the conditional entropy can be acheived using four or more messages 

[27].

Given a function of several variables, the communication complexity of computing in a distributed setting is considered in [35][16]. Early information theoretic work on communication complexity for distributed function computation includes [34]. In [28] the problem of computing at node- is considered and it is shown that bits are necessary and sufficient, where is a conditional entropy defined on the characteristic graph of , and . A characterization of the two-message rate region is also provided. More recently, two terminal interactive communication is studied in considerable detail in [21], [24], and the benefit of an unbounded number of messages is demonstrated. In particular tight bounds for computing the Boolean AND function are obtained.

If and

are iid Gaussian with unit variance, with correlation coefficient

, , the objective is to calculate at node-, and only a single round of communication is allowed from node- to node- then node- must send bits to achieve mean squared error distortion  [34]. If , the minimum rate required coincides with the rate for communicating with mse distortion , as can be seen from the rate distortion function for the source. If the objective is to determine at both nodes with mse distortion , the minimum sum rate is , which is twice the rate required for calculating at one node. Once again, when this coincides with the minimum rate for sending to and to , even if multiple rounds of interactive coding are allowed [30].

Our work is based on analysis techniques for quantization [4], [9],[36] some applications of which to detection problems have already appeared in [29][3] and  [11]. More recently, the design of fine scalar quantizers for distributed function computation with a squared error distortion measure is considered in [23] and succeeding works.

Ii Applications

While the problem considered in this paper is of fundamental importance, it also has potential applications to emerging systems for network security and machine learning.

The need for large scale distributed systems has been noted, by security researchers, as a foil to distributed and coordinated attacks. Examples of such attacks and pre-cursors to attacks are distributed denial of service attacks  [33], distributed port scans and fragmented worms. This is enabled by the increased sophistication of attackers, who are able to commandeer multiple resources and attack a network in a distributed manner, so as to evade localized detection techniques. A common feature of these attacks is that detection requires global information. The communication cost of detecting such attacks is high and the bottleneck is the network bandwidth, which is a few orders of magnitude smaller than memory bandwidth [2]. In response, researchers have considered the design of distributed, collaborative intrusion detection systems and several survey papers on this important subject have appeared recently, e.g [32][22].

A similar trend towards collaborative distributed systems is observed in the area of machine learning, e.g. [15]

. Machine learning systems can serve as a subsystem in an intrusion detection system, but are also of interest in a host of other applications. Primitives provided in such systems include gradient and stochastic gradient descent, map-reduce (for developing divide and conquer strategies) and graph parallel primitives. The problem of reducing the communications overhead in datacenter implementations of large scale machine learning problems has been addressed in several works, e.g. 

[19]

. As a specific example consider the design of a neural network classifier for data that is distributed across many physical locations 

[18]. The focus in  [18] is on understanding the statistical performance of the proposed distributed learning algorithm—there is no explicit accounting of the communication cost of the proposed algorithm. Our work aims to fill this gap.

Finally, we would like to note that when detecting an attack in a large network, the first attack is often very hard to detect. It is only after anomalous behavior is noted that an effort is made to discover the mode of the attack, after which an attack signature is obtained for preventing further spread. Thus, the availability of network data that precedes the attack is crucial for discovering the mode of an unseen attack. Such data can take the form of counts of packets with specific source, destination IP addresses, as in [13]. However, since uncompressed network logs will consume a lot of network bandwidth, it makes sense to have available a compressed representation of a network log. In the example of packet counts, lossy compression is an acceptable and necessary step in reducing the network bandwidth requirements. The emphasis on low delay is also important here. We may not have the luxury of accumulating data for a month at each sensor node, but may wish to encode data daily.

The problem considered here has applications to the above-mentioned distributed systems and our expectation is that the communication efficiencies obtained through their solution will contribute to system efficiencies.

Iii Lattice Setup

Fig. 1:

Voronoi region, Babai partition and three relevant vectors

Our analysis is for lattices in dimension 2. We summarize here, some of the necessary and relevant facts about two dimensional lattices. We will assume that generator matrix of lattice is of full rank (the associated lattice is called a full rank lattice) and has the upper triangular form

(1)

where the columns of are basis vectors for the lattice. The associated quadratic form is . It is known that this form is reduced if and only if and the three smallest values taken by over are , , and  see e.g. Th. II, Ch. II, [7]. Based on a result due to Voronoi, Th. 10, Ch. 21, [8], it follows that the relevant vectors, i.e. the vectors which determine the faces of the Voronoi cell, are , and . We thus consider lattices with generator matrix as above, with . From an additional symmetry, and in order to avoid indeterminate solutions we restrict such that . Performance at the endpoints and can be obtained by taking limits. More generally, the generator matrix of the lattice is represented by matrix with th column , . Thus . The entry of V is , thus . The Voronoi cell is defined as the set of all for which is the closest lattice point. When , we will write as shorthand for .

A fundamental region of a lattice is a set with the property that distinct points in the set are distinct, modulo translations by lattice vectors. The volume of any fundamental region is . The Voronoi cell is a fundamental region for the lattice . For the lattice with generator matrix (1), it is not hard to show that any translate of the rectangle is also a fundamental region for and that these are the only axis-aligned rectangular fundamental regions for this lattice. Given a lattice and a rectangular fundamental region , a partition of of the form , will be referred to as a rectangular fundamental partition of . A simple method for obtaining a fundamental rectangular partition is to partition into rectangles for which is constant, where

(2)

( is the nearest integer to ). This partition is referred to as the nearest-plane or Babai partition, the lattice point

(3)

is referred to as the Babai point, the set of mapped to by (2) and (3) is called the Babai cell associated with , denoted . The Babai cell at the origin is abbreviated .

Iv Interactive, Nearest-Plane, Single Round of Communication

Fig. 2: Two stages in the computation of . At the end of Stage-I, both nodes have determined . Stage-II then refines the approximation. Shown here is the 12 order for Stage-II communication. In the 21 order for Stage-II communication, is sent before and is calculated differently.

Our algorithm computes in two stages, as illustrated in Fig. 2. In the first stage, a Babai partition of is constructed. This is accomplished by first sending from node-2 to node-1 and then sending from node-1 to node-2 calculated according to (2). At the conclusion of this stage of the protocol, both nodes have determined an approximate nearest lattice point, , thus localizing to the Babai cell . In the second stage, we allow only a single round of communication. This round consists of sending a bin index from node-1 to node-2 and another bin index from node-2 to node-1. Computation of and ’s is explained later in this section. Different results are obtained depending on the order in which the are communicated. Both possibilities are analyzed. At the end of the second stage, each node has determined a better approximation to than . We call this common lattice point 333Our protocol excludes the possibility that the lattice points determined by each node at the end the the second stage are different..

Let and let denote the error probability. The total number of bits communicated in Stages I and II is denoted by , , respectively and . Our objective is to determine the error probability as a function of .

Since and are assumed to be independent and the Babai cell satisfies , the pdf of conditioned on the event or equivalently satisfies .

Iv-a Analysis of Rate in Stage-I

The Stage-I rate is given by , where and the are obtained in (2). For general probability distributions must be obtained computationally. However, when the lattice is scaled by (i.e. the generator matrix for the lattice is ) and it is easy to show that the Stage-I rate satisfies [5]

(4)

Iv-B Analysis: Stage-II, 12 Order

Fig. 3: A typical vertical strip created by and its partition into three parts by (left). Probability distribution which underlies the calculation of is on the right.

We now describe the scheme for the order for the second stage. Node-1 partitions into bins and sends a message to node-1 to indicate which bin lies in, in effect partitioning into vertical rectangular strips. Node-2 partitions each vertical rectangular strip into at most three parts using at most two horizontal cuts or thresholds444We shall assume that a vertical strip never straddles a vertical wall of a Voronoi cell.. The location of each cut is determined by the location of the appropriate boundary wall of a Voronoi cell. A typical situation is illustrated in Fig. 3. Here a rectangle is intersected by the boundary lines of the Voronoi cell , and is partitioned into three smaller rectangles. The partitioning of a rectangle into smaller rectangles is determined by the optimum decoding or decision rule, which associates a lattice point with every rectangle in the final partition. Consider a rectangle and let be the lattice point that it is decoded to. From elementary considerations it follows that . Thus the optimum decision rule decodes region to the lattice point whose Voronoi region has the largest probability of intersection with .

Assume that the upper and lower boundary lines of are described by and , respectively, when . In case one or more of the Voronoi boundary lines is absent, and coincide with the boundary of the Babai cell and the slopes are zero. Let have width and let and , both in , be as shown in Fig. 3. Then

(5)

and equality holds when . This determines the best location of two horizontal cuts for each vertical rectangular strip.

We now describe the partition of in a hierarchical manner, using random variables and . The function is constant on sub-intervals555Some intervals may be of zero length. For the cases that our analysis is applied to is either or . of , for some . describes the sub-interval that lies in. The sub-interval indexed by is special in that is zero over this sub-interval and for each . No further partitioning of a sub-interval is required, and Stage-II communication ends. Each subinterval is further partitioned into bins and the random variable describes the bin index of the bin that lies in. Thus each bin is indexed by , where is the index of the Babai cell, is the sub-interval index, is the bin index relative to the sub-interval, and

(6)

where is given by averaging the minimum value achieved by (5) with appropriate bin size .

The information rate from node-1 in Stage-II is . The sum information rate for all communication (Stages I and II) is given by

(7)

where, for convenience, we mention again that and are the random variables associated with communication in Stage-I, given by (2) and , and are random variables associated with communication in Stage-II.

Since the quantization in Stage-II is assumed to be fine (for a lattice at any scale), we can obtain useful approximations for . Specifically, for a realization of

(8)

where , the number of bins and , the th bin of the th sub-interval of the Babai cell indexed by . Note that may depend on though this is not reflected in the notation.

The partition constructed by node-2 is described next. Suppose lies in the rectangle . As shown in Fig. 3, is partitioned into at most three rectangles labeled , . Define the probability distribution , where . Node-2 sends bits to node-1 where

(9)

where lies in the th bin of the th sub-interval of the Babai cell indexed by , having length .

In order to derive limiting expressions for (5)–(9), we follow the approach in [4], [9],[36] and introduce the bin-length function and the point density function , where is the number of bins that a sub-interval is partitioned into and is the length of a bin that contains . Observe that measures the density of bins at within a sub-interval and integrates to unity over that sub-interval. Wherever needed will be indexed by the Babai cell index and sub-interval .

In terms of the point density function and

(10)

we obtain

(11)
(12)

and

(13)

Observe that (13) does not depend on .

We minimize with respect to the sum rate in two steps. First we obtain a lower bound on (11) through an application of Jensen’s inequality [12],  [36][10]. We follow this with another application of Jensen’s inequality to determine the optimal rate allocation . Thus for

and equality holds if and only if , for some constant . Let

(15)

and let

(16)

From (6) it follows that

and equality holds when for some constant . From here it follows directly that

(18)

where is the expectation with respect to the probability distribution defined in (16) and is the differential entropy of the probability density .

Remark 1.

From (6) we see that in order to achieve it is necessary that for all which have positive probability. This is impossible unless the bin size (the dimension) is zero, which requires an infinite rate.

Remark 2.

Conditions for convergence in (18) are less stringent than those required in the analysis of quantizers under difference distortion measures since the error measure considered here is the error probability. It suffices to assume that the marginal pdf’s satisfy smoothness conditions (53(a)-(c)) in [9].

Iv-C Special Case: 12 Order and

Fig. 4: Babai and Voronoi cells, with key points labeled. are the horizontal, vertical coordinates, resp. We have reduced the clutter by labeling only one of a pair of vertices , .

We now specialize the analysis to the simplest case where we assume that is uniformly distributed over . Thus in (18) and the remaining terms are derived in the sequel. We note here that this analysis also applies in a limiting sense when applied to lattice and . The only modification required is that be computed using (4). Thus the analysis presented here is applicable in the limiting case for general source distributions. The benefit is that it allows us to study explicity, the dependence on geometric parameters of the Babai and Voronoi cell.

The Voronoi cell and Babai cell are shown with all the significant boundary points and intervals in Fig. 4. We identify four thresholds , , and and five intervals , , , and with lengths , , , . Let . Note that in Fig. 4. Let . Note that in Fig. 4. Let the height of the Babai cell be . Thus

Then

(19)
(20)

and

(21)

Let random variable which takes values with probability . We thus obtain

(22)
(23)

It is worth noting that for the special case considered here, namely, uniformly distributed on , (23) can be obtained more directly by partitioning into equal-length intervals, into equal-length intervals, into interval, into equal-length intervals and into equal-length intervals [31].

Iv-D Interactive: Single Round, Reversed Steps,

Analysis is now presented for order of communication. The general formula (23) continues to apply here, but with , and replacing , , and , respectively. We thus derive an expression for the special case with uniformly distributed on , since this captures the essential geometric differences between the two orderings of communication in Stage-II.

The support for is partitioned into subintervals , and and the bin that lies in is communicated to node-1 by random variable . Conditioned on , random variable indicates a bin for interval , that lies in. Also let , the vertical () dimension of and , as in Fig. 4.

Node-2 sends , the index of the bin that lies in, and thus partitions into horizontal strips. Node-1 then partitions each horizontal strip into at most three parts using at most two vertical cuts or thresholds, referred to as the left and right thresholds, and sends to node-2. For a given , let be the probability that lies to the left of the left threshold (), the probability that lies to the right of the right threshold () and the probability that lies in between the two thresholds (). Let . Then

(24)

It follows that

(25)

Iv-E The Optimum Offset

Fig. 5: Shifted Babai cell (red) and Voronoi cells (blue) and lattice points ’+’. Observe that the lattice is not shifted and a lattice point remains at the center of every Voronoi cell. Also, the lattice point lies in the shifted Babai cell because of our restriction on . A single Babai cell will intersect at most six Voronoi cells.
Fig. 6: An illustration of the Voronoi cell an offset Babai cell (dashed rectangle) and the Babai cell with zero offset (solid rectangle).

We consider the possibility that the Babai partition constructed on for some fixed offset vector might lead to improved performance. Notice that the lattice and Voronoi partition remain unchanged; only the rectangular partition has shifted. It suffices to restrict to the rectangle and with this restriction . For 2D lattice considered here .

(a) (b) (c)
Fig. 7: Variation with offset for and bits for 12 order of communication. (a) as a function of , (b) given in (25) as a function of shows that is optimum. (c) 21 order of communication. is minimum at zero offset.

First consider the 12 order to communication. We have already shown that the error probability decreases as and thus the maximum rate of decay is obtained by choosing an offset for which is maximized. In terms of the distances shown in Fig. 6, and depend on the vertical offset . For , , and . Note that offset corresponds to . is maximized for any which satisfies , as shown in Fig. 7. Thus is optimal in terms of rate of decay. A further optimization is possible in terms of the constant term. This has been calculated numerically and is shown in Fig. 7(b). For the 12 order of communication is optimum (for all in ) and results in significant improvements in the error probability.

A similar, but simpler analysis for the reverse order shows that in this case the zero offset is indeed optimal, as shown in Fig. 7(c).

V Interactive: Infinite Rounds

Fig. 8: Red solid lines show partition after the first round of communication. Dashed lines are created in the second round of communication.

We now analyze the interactive model in which an infinite number of communication rounds are allowed. We provide an analysis under the assumption that is uniformly distributed over . As we have noted earlier, the analysis with this restriction also applies when the lattice is fine, i.e. when , the volume of a fundamental region is small. The construction and performance analysis is presented is described in Sec. V-A. The optimum offset is investigated in Sec. V-B.

V-a Interactive:Infinite rounds: A Bit-Excahnge Protocol

Node-2 communicates first. In Round-1, Node-2 partitions the support of into three intervals as in Sec. IV-D (see Figs. 8 and 4), , and , and . Let random variable be the index of the interval in which lies. In Round-1, upon receiving and if , Node-1 partitions the support of into three intervals , and (see Fig. 4). If , the support of is partitioned into intervals . If , no partitioning step is taken. Random variable describes the interval in which lies. Let , . Let , . Let and .

We assume that for every round, upon sending , Node- updates by subtracting the lower endpoint of the interval that it lies in.

The partition of into rectangular cells after a single, and after two rounds of communication is shown in Fig. 8. Define a rectangular cell to be error-free if its interior does not contain a boundary of . Of the seven rectangles in the partition at the conclusion of Round-1, all but four are error-free. If lies in an error-free rectangle, communication halts after Round-1. Else a second round of communication occurs, during which a total of 2 bits are communicated. This process of partitioning and communication continues until each node determines that lies in an error free rectangle of the current partition. When the algorithm halts, . Let , denote the number of rounds, and number of bits communicated, respectively, when the algorithm halts. Let and denote averages over .

Theorem 1.

For the interactive model with unlimited rounds of communication, a nearest plane partition can be transformed into the Voronoi partition using, on average, a finite number of bits and rounds of communication. Specifically,

(26)

and

Proof.

We assume that an optimum entropy code is used (thus if , the codeword length is bits). The term in (26) is the cost of resolving the Round-1 partition. At the conclusion of Round-1, if belongs to a region which is not error-free, then the average number of bits transmitted is obtained by the following argument. At the conclusion of Round-1, there are two kinds of error rectangles, determined by the sign of the slope of the boundary of in the rectangle. Note that error rectangles are designed so that the boundary of is a diagonal of the corresponding rectangle. Let an error rectangle have length and height . If the slope is positive, construct the binary expansion